Irregular Polygon Calculator

What is the Polygon Area Calculator?

The Polygon Area Calculator is an interactive web application that calculates essential properties of regular polygons – shapes with equal sides and equal angles. Unlike basic calculators that only determine area, your tool provides seven key parameters:

• Side length (a): The length of each equal side of the polygon
• Inradius (r): Radius of the inscribed circle (apothem)
• Circumradius (R): Radius of the circumscribed circle
• Area (A): The total surface area enclosed by the polygon
• Perimeter (P): The total distance around the polygon
• Interior Angle: The angle between adjacent sides inside the polygon
• Exterior Angle: The angle formed when extending one side outward.

How to Use the Calculator

Selecting Polygon Type and Units

1. Choose Number of Sides: Select from common polygons (3-12 sides) or input less common options (15, 20, 24, 30, 50, or 100 sides).
2. Select Measurement Units: Choose from metric (mm, cm, m, km) or imperial units (in, ft, yd) based on your preference and location.
3. Input methods: Our calculator offers five flexible input approaches through a tabbed interface.

How the Calculator Works

The calculator uses trigonometric relationships specific to regular polygons to compute all parameters from any single known value. The mathematical foundation is based on dividing the polygon into congruent isosceles triangles and applying trigonometric functions.

Core Mathematical Relationships

The calculator maintains these fundamental relationships for all calculations:

• Perimeter: P=n×aP=n×a (number of sides × side length)
• Interior Angle: θint=(n−2)×180∘nθint​=n(n−2)×180∘​
• Exterior Angle: θext=360∘nθext​=n360∘​
• Relationship between radii and side: R2=r2+(a2)2R2=r2+(2a​)2.

Calculation Methods by Input Type

1. From Side Length (a)

When side length is known, the calculator determines:

• Inradius: r=a2×tan⁡(π/n)r=2×tan(π/n)a​ 
• Circumradius: R=a2×sin⁡(π/n)R=2×sin(π/n)a​ 
• Area: A=n×a24×tan⁡(π/n)A=4×tan(π/n)n×a2​ or A=n×a×r2A=2n×a×r​ 

2. From Inradius (r)

When apothem is known:

• Side length: a=2×r×tan⁡(π/n)a=2×r×tan(π/n) 
• Area: A=n×r2×tan⁡(π/n)A=n×r2×tan(π/n) 

3. From Circumradius (R)

When circumradius is provided:

• Side length: a=2×R×sin⁡(π/n)a=2×R×sin(π/n) 
• Area: A=n2×R2×sin⁡(2π/n)A=2n​×R2×sin(2π/n) 

4. From Area (A)

When only area is known, the calculator works backward using area formulas to derive all other parameters.

5. From Perimeter (P)

When perimeter is provided, it first calculates side length (a=P/na=P/n) then applies the side length formulas.